Unified Flow Rule of Undeveloped and Fully Developed Dense Granular Flows Down Rough Inclines

Abstract

We report on chute measurements of the free-surface velocity $v$ in dense flows of spheres and diverse sands and spheres-sand mixtures down rough inclines. These and previous measurements are inconsistent with standard flow rules, in which the Froude number $v/\sqrt{gh}$ scales linearly with $h/h_s$ or $(\tan \theta /{\mu }_r{)}^{2}h/h_s$, where ${\mu }_r$ is the dynamic friction coefficient, $h$ the flow thickness, and $h_s(\theta )$ its smallest value that permits a steady, uniform dense flow state at a given inclination angle $\theta$. This is because the characteristic length $L$ a flow needs to fully develop can exceed the chute or travel length $l$ and because neither rule is universal for fully developed flows across granular materials. We use a dimensional analysis motivated by a recent unification of sediment transport to derive a flow rule that solves both problems in accordance with our and previous measurements: $v=v_{\infty }[1-\exp (-l/L){]}^{1/2}$, with $v_{\infty }\propto {\mu }_r^{3/2}{\left[(\tan \theta -{\mu }_r)h\right]}^{4/3}$ and $L\propto {\mu }_r^3{\left[(\tan \theta -{\mu }_r)h\right]}^{5/3}h$.

References in corpus (3)

• Relation between dry granular flow regimes and morphology of deposits: formation of levees in pyroclastic deposits
• Flow rule, self-channelization and levees in unconfined granular flows
• Rapid granular flows on a rough incline: phase diagram, gas transition, and effects of air drag